Question 3
Let $n \ge 3$ be a natural number, and let $\mathcal{F}$ be a family consisting of at most $n$ distinct subsets of the set $\{1, 2, \dots, n\}$ with the property that one can consider $n$ distinct points in the plane, labeled with the numbers $1, 2, \dots, n$, then draw segments between some of these points such that, for any distinct numbers $i, j \in \{1, 2, \dots, n\}$, the points labeled $i$ and $j$ are connected by a segment if and only if the number $i$ belongs to exactly $j$ subsets in $\mathcal{F}$. Find the maximum possible value of the sum of the number of elements of the sets in $\mathcal{F}$.